Design and Performance Investigation of Control Charts Based on Ordinal Categorical Data

Abstract

In real life, consumers naturally prefer the products or service with high quality especially when their consumption habitat is upgrading. As a result, in either manufacturing or service industry, the enterprises tend to regard the quality as one of the most important factors to gain competitive advantages. Generally speaking, the quality of products or service is inversely proportional to the variability of associated process. Although part of the process variability is invoked by unavoidable causes, statistical approaches such as statistical process control (SPC) are quite useful for distinguishing assignable causes from those unavoidable and further reducing variability. Among all the SPC techniques, control charts are prevalent for either analytical purpose mainly corresponding to Phase I change-point detection or for controlling purpose concerned with Phase II online monitoring. Up to now, tremendous control charts have been developed for quality improvement or quality control in various fields.

Based on the nature of quality characteristics, we can classify control charts into two types, those with continuous data and those with categorical data. Whether the process involves one continuous characteristic or multiple ones, sufficient charting techniques are available for change-point detection in Phase I or online surveillance in Phase II. However, the same cannot be said for processes with one or more categorical quality characteristics. Furthermore, the few existing categorical control charts neglect the order within attribute levels of such characteristics so that they may not perform satisfactorily well in such cases. Regarding this, we presume that a latent continuous variable underlies the observable ordinal factor and certain predefined thresholds cut the range of this variable into such intervals that exhibit one-to-one relationship with attribute levels of the corresponding ordinal factor. Based on this assumption, the work introduces average cumulative probability, modified log-linear model, standardized ranks and spatial signs to incorporate associated ordinal information in Phase I or Phase II analysis of processes respectively involving one ordinal factor, multiple factors, and mixed ordinal and continuous factors.

The first part proposes systematic methods for simultaneous surveillance of the location and dispersion of the continuous variable underlying an ordinal categorical factor, as well as detecting the change point in the location of the latent continuous variable. This part overcomes the drawbacks of existing methods in neglecting online surveillance of dispersion shifts and ordinal information among attribute levels of an ordinal factor for change-point detection. To be specific, when probabilities corresponding to ordinal levels are known in advance or have been estimated from IC dataset, we prove that the average cumulative probability is equivalent to the standardized rank for each attribute level. Then the charting statistic is designed based on the deviation of observed average cumulative level counts from their expectations calculated from average cumulative probabilities. Simulation results show that the subsequent control chart is omnipotent with regard to simultaneously monitoring location and dispersion shifts of latent continuous variable while its competitor only focuses on detecting the location ones.

When the IC parameters are unknown, the collected dataset will receive retrospective analysis to eliminate the unusual patterns and change-point detection is one of the important tools to complete this job. In our work, the log-linear model with one categorical factor is integrated with certain quantity obtained under null hypothesis to reflect the ordinal information among the attribute levels. Then at-most-one-change-point(AMOC) framework is introduced to derive the statistic for detecting the latent change point in ordinal categorical processes. Simulation study demonstrates that the proposed change-point detection scheme is advantageous over that without account of ordinal information. For the purpose of exposition, the proposed two techniques are utilized to monitor location and dispersion of latent continuous variable and detect latent change point in a 3D printing process.

The second part tackles monitoring problems with multiple quality factors, each of which is classified with ordinal attribute levels. This part is advantageous in characterizing the ordinal information among attribute levels of each factor into an extended log-linear model for multivariate ordinal categorical process control while previous control charts neglect such ordinal information. Base on the first part, it is straightforward to extend the modified log-linear model with one factor to its multivariate version. Then we introduce the linear-by-linear (L×L) association framework to represent the dependence between any pair of ordinal factors without neglecting the ordinal information. The finalized log-linear model reflects both marginal distribution of ordinal factors and the two-factor interaction effects with ordinal information in each categorical factor. Generalized likelihood ratio test (GLRT) method is exploited to develop a closed-form charting statistic. Compared with the most recent method in simulation study, the proposed multivariate categorical control chart exhibits superior performance in detecting location shifts and dependence shifts in the corresponding latent continuous variables of ordinal categorical characteristics merely with the attribute-level counts of such characteristics. Finally, the example of standard test units manufactured by 3D printer is introduced to display the use of proposed control chart.

The third part copes with the situation where several ordinal categorical characteristics are combined with multiple real continuous characteristics to induce mixed-type data. Despite the fact that limited statistical approaches are developed to characterize or even monitor the relationships among them, this part overcomes the difficulty in establishing a unified framework between categorical data and continuous data. Specifically, we notice that an order exists not only among continuous observations but also among ordinal categorical observations so that it is reasonable to ranking these observations to transform them into a unified framework of standardized ranks. Then the spatial signs are combined with the standardized ranks to derive spatial-sign covariance matrix for statistical surveillance. Such appropriateness relies in the fact that standardized ranks follow the uniform distribution with parameter 0 and 1 so as to form a directionally symmetric space. It allows spatial signs to utilize the directions of observations with regard to their common center instead of magnitudes of these observations and. Finally, charting statistic is derived based on testing if spatial-sign covariance matrix deviates its expectation, i.e., identity matrix and its affine invariant property guarantees the high efficiency of detecting dependence shifts among orginal mixed-data. Simulation study demonstrates the advantages of proposed control chart over its competitor and the ease of implementation is displayed by a practical example.

Date of Award	12 Jul 2018
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Min XIE (Supervisor) & Qin Su (External Supervisor)